Exact equations for smoothed Wigner transforms and homogenization of wave propagation
Agissilaos Athanassoulis, K. S. Politis · PAMM · 2007
Abstract The Wigner transform (WT) is a quadratic transform that takes an oscillatory function u (x): ℝn ↦ ℂd to a phase‐space density W (x, k) = W [u ](x, k): ℝ2n ↦ ℂd ×d , resolving it over an additional set of 'wavenumber' variables. The WT and its variations have been heavily used in quantum mechanics, semiconductors, homogenization of wave equations, timefrequency analysis, signal processing, pseudodifferential operators etc. The WT however has a fundamental difficulty: WTs exhibit artifacts, collectively known as ‘interference terms’, and can be arbitrarily more complicated than the original wavefunction. A very successful, well established way to go around this is using the Wigner measures (WMs), a semiclassical approximation to the WT. We propose a different approach, namely smoothing the WT with an appropriate kernel. Such smoothed WTs (SWTs) have been used with great success in signal processing. They have not been used in the treatment of PDEs, a fundamental obstacle being the lack of exact equations governing their evolution. We present the machinery which allows the coarse‐scale reformulation of a broad class of wave problems in terms of the SWT, along with numerical experiments which clearly show the validity and applicability of the method. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)